A Course on Borel Sets

The roots of Borel sets go back to the work of Baire [8]. He was trying to come to grips with the abstract notion of a function introduced by Dirich­ let and Riemann. According to them, a function was to be an arbitrary correspondence between objects without giving any method or procedure by which the correspondence could be established. Since all the specific functions that one studied were determined by simple analytic expressions, Baire delineated those functions that can be constructed starting from con­ tinuous functions and iterating the operation 0/ pointwise limit on a se­ quence 0/ functions. These functions are now known as Baire functions. Lebesgue [65] and Borel [19] continued this work. In [19], Borel sets were defined for the first time. In his paper, Lebesgue made a systematic study of Baire functions and introduced many tools and techniques that are used even today. Among other results, he showed that Borel functions coincide with Baire functions. The study of Borel sets got an impetus from an error in Lebesgue's paper, which was spotted by Souslin. Lebesgue was trying to prove the following: Suppose / : )R2 -- R is a Baire function such that for every x, the equation /(x,y) = 0 has a. unique solution. Then y as a function 0/ x defined by the above equation is Baire.

Contenu

1 Cardinal and Ordinal Numbers.- 1.1 Countable Sets.- 1.2 Order of Infinity.- 1.3 The Axiom of Choice.- 1.4 More on Equinumerosity.- 1.5 Arithmetic of Cardinal Numbers.- 1.6 Well-Ordered Sets.- 1.7 Transfinite Induction.- 1.8 Ordinal Numbers.- 1.9 Alephs.- 1.10 Trees.- 1.11 Induction on Trees.- 1.12 The Souslin Operation.- 1.13 Idempotence of the Souslin Operation.- 2 Topological Preliminaries.- 2.1 Metric Spaces.- 2.2 Polish Spaces.- 2.3 Compact Metric Spaces.- 2.4 More Examples.- 2.5 The Baire Category Theorem.- 2.6 Transfer Theorems.- 3 Standard Borel Spaces.- 3.1 Measurable Sets and Functions.- 3.2 Borel-Generated Topologies.- 3.3 The Borel Isomorphism Theorem.- 3.4 Measures.- 3.5 Category.- 3.6 Borel Pointclasses.- 4 Analytic and Coanalytic Sets.- 4.1 Projective Sets.- 4.2 ?11 and ?11 Complete Sets.- 4.3 Regularity Properties.- 4.4 The First Separation Theorem.- 4.5 One-to-One Borel Functions.- 4.6 The Generalized First Separation Theorem.- 4.7 Borel Sets with Compact Sections.- 4.8 Polish Groups.- 4.9 Reduction Theorems.- 4.10 Choquet Capacitability Theorem.- 4.11 The Second Separation Theorem.- 4.12 Countable-to-One Borel Functions.- 5 Selection and Uniformization Theorems.- 5.1 Preliminaries.- 5.2 Kuratowski and Ryll-Nardzewski's Theorem.- 5.3 Dubins - Savage Selection Theorems.- 5.4 Partitions into Closed Sets.- 5.5 Von Neumann's Theorem.- 5.6 A Selection Theorem for Group Actions.- 5.7 Borel Sets with Small Sections.- 5.8 Borel Sets with Large Sections.- 5.9 Partitions into G? Sets.- 5.10 Reflection Phenomenon.- 5.11 Complementation in Borel Structures.- 5.12 Borel Sets with ?-Compact Sections.- 5.13 Topological Vaught Conjecture.- 5.14 Uniformizing Coanalytic Sets.- References.